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\section{Discussion}

Our calibration dataset (dataset \#1) contained detailed volume measurements for 8192 trees from 19 species, that can be considered as a very large database in comparison with the literature. Most references about expansion factors found in the literature were based on volume or biomass estimations obtained through allometric equations and were not directly based on field measurements. Modeling $VEF$ directly from field measurements enabled to avoid propagation of modeling errors.

The data used for calibrating the models were collected between 1920 and 1985. It is not impossible that the shape of trees has changed since these measurements due to changes in silvicultural practices or, less likely, to the climate change effect. However, the date of measurements is probably not the major source of variation in our database. 

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The major weakness of the calibration dataset is that it represents at 90\% pure, even-aged high forests \cite{Vallet2006} and rather small and medium trees in diameter compared to the French National Forest Inventory data (not shown). 
For \textit{Fraxinus} species, the model was calibrated with trees ranging between 26.0 and 136.5 cm in diameter whereas five trees from dataset \#2 had $C130$ ranging between 140.5 and 204.0 cm. The model was therefore used in extrapolation for these trees leading to big errors both in terms on $VEF$ and $Vtot$. 
For \textit{Pseudotsuga}, the trees from the calibration dataset were young with a maximum $C130$ of 107 cm since the species was recently introduced in France at the date of measurements. It would therefore be preferable to use the generic gymnosperm model (i.e., fixed effects and $G = 0$) for this species.
For \textit{Fagus} and \textit{Quercus} species, a quite high variability of $VEF$ was observed, in particular for trees belonging to the calibration dataset. This variability could be explained by differences in origin of the trees (e.g., region, site, silviculture). For both species, the big trees of dataset \#2 coming from coppice-with-standards stands tended to have underestimated values of $VEF$, except for one declining tree. 
The presence of a fork had a big impact on $VEF$. For a given $C130$, a forked tree had a bigger woody volume than a not forked tree. This behaviour was observed in our study for \textit{Quercus robur/petraea} and \textit{Quercus pubescens}. 
It is important to remind that dataset \#2 was specially chosen (see Section \ref{M_and_M}) to complement the available data with under-represented species, regions or $C130$ values, which often lead to use the model in extrapolation for trees belonging to this dataset.     
The predictions obtained for trees consistent with the calibration dataset (i.e., high forests, same range of $C130$, not forked trees) were totally satisfactory. To go further, we tested a calibration of the model based on dataset \#1 and \#2 together since the two datasets are complementary. However, dataset \#2 was two small in comparison with dataset \#1 to really impact significantly the parameter estimates. In the model with $C130$ only it was observed that the slopes were slightly increasing for \textit{Fraxinus} and for \textit{Quercus}, which tended to improve the model for big trees.  

It was not possible to get easily a reliable information about the silviculture for each tree of the dataset \#1 and therefore to take this aspect into account in our models. However, it is known that the slenderness coefficient (i.e., height over diameter at breast height ratio) depends on the silviculture through the effect of the competition between trees \citep{Jagodzinski2009}. This silvicultural effect was at least partially, but not perfectly, taken into account in our model through the variable $\frac{C130}{H^{2}}$. It was particularly interesting to notice that this variable was the square of the \textit{hardiness} used by \cite{Vallet2006} to model $Vtot$ directly from $C130$ and height. 

Regarding the parameter estimates, the model seemed to be consistent at several levels. The value of 21.447 for $\beta_1$ (20.231 in the model with $C130$ only), which controls the curve location along the X-axis, was in the order of the circumference limit of $7 \cdot \pi$ cm that was fixed in the study. The error variance power values $\delta_{G_{i}}$ obtained for angiosperms and gymnosperms were negative, which was consistent with an increasing of the $VEF$ variability for small diameter trees. It was shown that the variability was higher for angiosperms than for gymnosperms, which reflected a greater variability in tree shape for angiosperms, probably related to the silvicultural practices. The slopes of the relationship between $VEF$ and $C130$ were much higher for angiosperms than for gymnosperms indicating that for a same volume in the stem, angiosperms had much more volume in their branches.   

The species-specific behaviour was especially visible for relatively big trees. The slope of the relationship between $VEF$ and $C130$ was almost null for \textit{Picea} and \textit{Abies} species, and relatively low for all gymnosperms, except \textit{Pseudotsuga}. The slopes were stronger for angiosperms with the lowest slope for \textit{Quercus} and the highest slope for \textit{Carpinus}. For a medium tree of 100 cm in circumference, the $VEF$ computed with the model represented in Fig. \ref{Predicted_VEF_vs_genus} were 1.066 for \textit{Picea}, 1.078 for \textit{Abies}, 1.106 for \textit{Larix}, 1.137 for \textit{Pinus}, 1.189 for \textit{Quercus}, 1.209 for \textit{Pseudotsuga}, 1.252 for \textit{Betula}, 1.309 for \textit{Fagus}, 1.343 for \textit{Fraxinus} and 1.492 for \textit{Carpinus}. These differences probably reflect differences in tree architecture and possibly in the silviculture traditionally associated with the considered species. Regarding our results and in particular Fig. \ref{Predicted_VEF_vs_genus}, the approximate value of 1.5 given by \cite{Pretzsch2009} seems appropriate for angiosperms but not for gymnosperms for which the $VEF$ and $Vtot$ would be highly overestimated. Some group of species appeared like \textit{Picea} and \textit{Abies} or like \textit{Fraxinus} and \textit{Fagus} and in a further study it would be interesting to link these groups to architectural and ecological traits known for these species.

Regarding the \textit{Pinus} example, many reasons could explain the observed differences between the seven \textit{Pinus} species in term of $VEF$. \textit{Pinus sylvestris} trees having small diameter and coming mainly from the same forest area were particularly slender, i.e., with a long part of the stem less than 7 cm in diameter, which might result in high expansion factors for these very small $C130$. At the opposite, for bigger $C130$, \textit{Pinus pinaster} trees showed higher expansion factors than \textit{Pinus sylvestris} trees. It was very difficult to say if it was a species effect only, or rather a site or silvicultural effect. This also raises the question of developing models at the species level (for each \textit{Pinus} and \textit{Quercus} species for example) rather than at the genus level.

Databases and publications exist that provide wood basic density for numerous temperate tree species, which could be used to convert easily $VEF$ to $BCEF$. It was particularly interesting to compare the shape of the plots of $VEF$ as a function of $C130$ to the equivalent plots found in the literature for $BEF$ or $BCEF$. In our model the slope of the relationship was constrained to be positive for big trees, whichever the dataset considered. In the literature, the slopes obtained for $BEF$ were often negative \citep{Somogyi2007,Pajtik2008,Pajtik2011,Sanquetta2011} because the trees were still young or small, and only the first part of the relationship (i.e., the decreasing part) was obtained. Plots of $BEF$ as a function of growing stock at the stand level (in m$^3$/ha) show negative slopes as well \citep{Brown2002,Lehtonen2004,FAO2005,Guo2010}. The latter emphasizes that our dataset including big trees was particularly valuable.




